Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. Proposed by Amine Natik, Morocco

Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$. Show that $FG$ is perpendicular to $IO$.

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If \[x^n-y^n=x^{n+1}-y^{n+1},\]then show that \[1<x+y<\frac{2n}{n+1}.\]

Given a connected graph with $n$ edges, where there are no parallel edges. For any two cycles $C,C'$ in the graph, define its outer cycle to be \[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\](1) Let $r$ be the largest postive integer so that we can choose $r$ cycles $C_1,C_2,\ldots,C_r$ and for all $1\leq k\leq r$ and $1\leq i$, $j_1,j_2,\ldots,j_k\leq r$, we have \[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\](Remark: There should have been an extra condition that either $j_1\neq i$ or $k\neq 1$) (2) Let $s$ be the largest positive integer so that we can choose $s$ edges that do not form a cycle. (Remark: A more precise way of saying this is that any nonempty subset of these $s$ edges does not form a cycle) Show that $r+s=n$. Note: A cycle is a set of edges of the form $\{A_iA_{i+1},1\leq i\leq n\}$ where $n\geq 3$, $A_1,A_2,\ldots,A_n$ are distinct vertices, and $A_{n+1}=A_1$.

Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of positive integers such that $a_0=0,a_1=1$, and \[ a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots. \]for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.

A calendar is a (finite) rectangular grid. A calendar is valid if it satisfies the following conditions: (i) Each square of the calendar is colored white or red, and there are exactly 10 red squares. (ii) Suppose that there are $N$ columns of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the top row to the bottom row, and within each row from left to right, there do not exist $N$ consecutive numbers such that the squares they are in are all white. (iii) Suppose that there are $M$ rows of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the left-most column to the right-most column, and within each column from bottom to top, there do not exist $M$ consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by $90^{\circ}$, the resulting calendar still satisfies (ii). How many different kinds of valid calendars are there? (Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the $10$ red squares are at different locations.)

Let $I$ be the incenter of triangle $ABC$, and $\ell$ be the perpendicular bisector of $AI$. Suppose that $P$ is on the circumcircle of triangle $ABC$, and line $AP$ and $\ell$ intersect at point $Q$. Point $R$ is on $\ell$ such that $\angle IPR = 90^{\circ}$.Suppose that line $IQ$ and the midsegment of $ABC$ that is parallel to $BC$ intersect at $M$. Show that $\angle AMR = 90^{\circ}$ (Note: In a triangle, a line connecting two midpoints is called a midsegment.)

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$holds for all $ x,y \in \mathbb{N} $

For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$ Proposed by Alex Zhai, United States